We show that log flat torsors over a family $X/S$ of nodal curves under a finite flat commutative group scheme $G/S$ are classified by maps from the Cartier dual of G to the log Jacobian of X. We deduce that fppf torsors on the smooth fibres of $X/S$ can be extended to global log flat torsors under some regularity hypotheses.
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Mehidi et al. (2024) studied this question.
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